Diffusion Models for SU(2) Lattice Gauge Theory in Two Dimensions

TL;DR

This paper demonstrates score-based diffusion models for sampling two-dimensional SU(2) lattice gauge configurations with the Wilson action, addressing non-Abelian geometry via a quaternion representation. The authors train on $10^4$ Hybrid Monte Carlo configurations at $β_0=2.0$ on an $8\times8$ lattice (augmented to $2\times10^4$ by random gauge transformations) and use a fully convolutional U-Net to perform physics-conditioned diffusion, enabling generation at other couplings without retraining. Validation against exact finite-volume plaquette predictions shows biases $|Δ|$ as small as $10^{-3}$ near the training coupling and modest deviations up to $O(10^{-2})$–$O(10^{-1})$ at larger volumes or couplings, with action densities generally matching the analytical result $S/V=-β\,I_2(β)/I_1(β)$. The work confirms diffusion models as a viable tool for non-Abelian gauge-field generation in low dimensions and motivates extensions to higher-dimensional theories and more complex gauge groups, including cross-method comparisons with gauge-equivariant diffusion frameworks.

Abstract

We apply score-based diffusion models to two-dimensional SU(2) lattice pure gauge theory with the Wilson action, extending recent work on U(1) gauge theories. The SU(2) manifold structure is handled through a quaternion parameterization. The model is trained on 10,000 configurations generated via Hybrid Monte Carlo at a fixed coupling $β_0= 2.0$ on an $8\times 8$ lattice, augmented to 20,000 samples via random gauge transformations. Through physics-conditioned sampling exploiting the linear $β$-dependence of the score function, we generate configurations at different values of the coupling without retraining; through the fully convolutional U-Net architecture with periodic boundary conditions, we generate configurations on lattices of different spatial extents. We validate our approach by comparing the average plaquette and Wilson action density against exact analytical predictions. At the training lattice size ($8\times 8$), the model reproduces the exact plaquette with biases $|Δ| \leq 0.001$ for $β\in [1.5, 2.5]$ and $|Δ| < 0.06$ across $β\in [1, 4]$. For lattices sharing the training extent $L=8$ in at least one direction, biases remain below $\sim 0.003$ for $β\in [1.5, 2.5]$, with larger deviations at higher couplings. This work demonstrates that diffusion models are a promising tool for non-Abelian gauge field generation and motivates further investigation toward higher-dimensional theories.

Figures (12)

• Figure 1: Complete pipeline of the diffusion model approach. Training data is generated via HMC at coupling $\beta_0$, augmented through random gauge transformations, and used to train the diffusion model. Once trained, the model generates new configurations at arbitrary coupling $\beta$ and lattice extents without retraining.
• Figure 2: Detailed architecture of the U-Net neural network $\epsilon_\theta$. The network takes the noised configuration $\varphi_t$ and timestep $t$ as inputs and predicts the noise $\epsilon_{\text{pred}}$. The encoder-decoder structure with skip connections enables multi-scale feature extraction while maintaining spatial resolution in the output.
• Figure 3: Flowchart of a single training step. The model learns to predict the noise $\epsilon_{\text{true}}$ that was added to clean configurations, enabling the reverse denoising process during sampling.
• Figure 4: Sampling algorithm flowchart. Starting from Gaussian noise, the iterative denoising loop applies physics-conditioned noise prediction and gradually recovers a gauge field configuration. The final projection step ensures the output satisfies the SU(2) constraint.
• Figure 5: Plaquette distribution from 500 diffusion-generated configurations at $\beta = 2.0$ on the $8 \times 8$ training lattice. Red dashed line: exact value $I_2(\beta)/I_1(\beta)$; blue line: sample mean. The distribution is well-centered on the exact result.